if (0,-7) lies on the terminal side of angle theta in standard position, find each value sec(theta) , cot(theta) and sin(theta)
(x,y) = (0,-7) so r=7
sec = r/x =7/0 = -oo
cot = x/y = ...
sin = y/r = ...
To find the values of sec(theta), cot(theta), and sin(theta) when the point (0, -7) lies on the terminal side of angle theta in standard position, we need to use the given information to determine the values of the trigonometric ratios.
When a point lies on the terminal side of an angle theta in standard position, we can use the Pythagorean theorem to find the hypotenuse and the values of sin(theta) and cos(theta).
In this case, the given point is (0, -7). Let's call the hypotenuse 'r'. Since the point lies on the y-axis, we can determine the value of r by using the y-coordinate of the point. Therefore, r = |-7| = 7.
Now, we can determine the value of sin(theta) by dividing the opposite side (which is -7) by the hypotenuse (which is 7). So, sin(theta) = -7/7 = -1.
Since sec(theta) is the reciprocal of cos(theta), we need to find the value of cos(theta) first. We know that cos(theta) = adjacent/hypotenuse. In this case, the adjacent side is 0 (as the point lies on the y-axis), and the hypotenuse is 7. So, cos(theta) = 0/7 = 0.
Now, we can find sec(theta) by taking the reciprocal of cos(theta). Therefore, sec(theta) = 1/cos(theta) = 1/0. Since division by zero is undefined, sec(theta) is also undefined.
Lastly, we can find cot(theta) by taking the reciprocal of tan(theta). Since tan(theta) = sin(theta)/cos(theta), and cos(theta) has a value of 0, tan(theta) is also undefined, and therefore cot(theta) is undefined.
To summarize,
- sin(theta) = -1
- sec(theta) = undefined
- cot(theta) = undefined